CoMPLEX
University College London
Mini Project 2
Freeze Drying of Medicine
Dimitrios Voulgarelis
Supervisors: Prof Frank Smith & Prof Ajoy Velayudhan
Abstract
In the current project a discussion is given regarding freeze-drying or lyophilization with
specific interest in its application to the storage of pharmaceutical products. Moreover, a
number of important mathematical and computational models in the field of freeze-drying
are presented, with the main focus being placed on their differences, uses, conclusions
and evolution from the first simple models to more sophisticated ones and back again to
simpler models for different purposes. Finally, the last part of the project is an original
mathematical approach for the description of the freezing stage of lyophilization, using
the Cauchy Integral formulation.
Contents
1 Lyophilization
2
2 Modelling The Freeze-Drying Process
2.1 Freeze-Drying in Trays . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Freeze-Drying in Vials . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
6
8
3 Mathematical Description of The Small Time Behaviour of The Freezing
Stage in Vials
3.1 Top and Side Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Top Corners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Bottom Corners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Full Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
12
13
16
17
A Latent Heat Boundary Condition After Conformal Mapping
21
B Leading Order Approximation of Integrals In The Two Limit Cases
23
C Asymptotic Limits of Θ, Ψ On The Corner’s Sides
25
1
Chapter 1
Lyophilization
Lyophilization or Freeze-Drying is defined as a stabilizing process [20] (removal of solvent) where initially the substance is frozen and then dried, first by sublimation and later
by desorption, until the moisture reaches low levels that prevent reactive and degradative processes. There are several advantages to this type of drying over others. The
most important and relevant to complex medicines (macromolecules) is that drying of
the solvent occurs at low temperature without excessive heating that could denature the
proteins. Further advantages include lower transportation costs since, removing solvent
means removing weight, reduced oxidative denaturation since the product is sealed in
low vacuum or inert gas, easy reconstitution by adding diluents [20], highly controlled
environment, increase of storage life as well as very low water content at the end of the
process. A different preservation method was freezing preservation which came with very
high transportation costs due to large weights and the possibility of total loss in the case
of malfunctioning of the freezing plant. Hence, both the advantages of freeze-drying and
the disadvantages of freeze preservation led a lot of people to the study and adoption of
the first. Pharmaceutical products are usually stored in vials with a stopper on top to
seal the product after the process.
Figure 1.1: Phase diagram showing the triple point threshold A. In order to avoid the
liquid state the partial water pressure must stay below that point [3].
2
The lyophilization process is constituted of three stages (Figure 1.2 [3]). The first
stage is the freezing of the product. During this stage the temperature of the product
is lowered in order to go from the liquid to the solid state. The type of freezing (rate of
freezing) can directly affect the product quality and is considered to be very important
since it affects the crystalline structure (size/shape of the crystals) of the frozen product
which in turn affects the drying stage. On the one hand, higher rates usually produce
small crystals, which prolong the drying times due to the high resistance of vapor flow
while lower rates produce large crystals that minimize the resistance to moisture [], but
on the other hand slow freezing can cause decrease of the pH of some forms of sodium
phosphate resulting in decreased stability of the frozen and freeze-dried drug and it can
also cause cold denaturation of certain proteins [17]. It is worth noticing that a few other
factors control the form of this structure, other than the freezing rate. Such factors are
the nature of the product as well as the processing before the freezing. Depending on
the nature of the product the freezing must proceed below a critical temperature. For
crystalline material that temperature is the Eutectic temperature whereas for an amorphous mix it is the Glass Transition temperature. In the case of an incomplete freezing
there is the danger of foaming (melt back) when low vacuum is applied and hence of
product deterioration [2]. During freezing not all water is removed but some remains as
bound/adsorbed water.
Following the complete freezing of the product, the second stage starts which is called
the primary drying stage. At this stage the pressure is dropped below the triple point,
as seen in Figure 1.1 [3], in order to avoid the liquid state and for sublimation to occur. Constant heating needs to be applied to the product, since sublimation causes
energy/temperature loss (latent heat loss). So, without sufficient heat the product will
cool which will result in an extension of the drying time. Heating control is vital during the primary drying stage for a number of reasons. Besides the prolonging of drying
times, due to insufficient heat, too much heat could mean melting (foaming) and hence
result in an unacceptable product. The general constraint is that the temperature of the
frozen product must not exceed the critical temperature mentioned before and although
this might not be too difficult for products stabilized by crystalline materials (eutectic
temperature is high), proteins and peptides require different excipients with much lower
critical temperatures. Another essential control parameter is the pressure. As already
mentioned, the partial water vapor pressure needs to be constantly maintained below the
triple point. Moreover, the partial pressure surrounding the product must be lower than
the partial pressure of the ice itself for evaporation to occur [2]. Hence, decreased pressure
leads to increased evaporation but below a certain value a further decrease in pressure
will not improve evaporation but will act as a barrier to heat transfer [2]. Primary drying
is the longest stage and a main reason behind that is the fact that the migration of the
vapor is a diffusion process. This stage ends when all ice has sublimed and only the
bound moisture remains. A small amount of bound water is removed during primary
drying but the majority is removed in the next stage.
The third and final stage of lyophilization is the secondary drying stage where the
chemically bound water is reduced to the desired level to enhance stability. After the end
of the primary drying stage the moisture is still at relatively high levels 7-8% [3]. During
3
the secondary drying the temperature of the product is raised, while staying again below
the constraints and the pressure is further dropped. The end product moisture levels are
usually between 1-3%.
Figure 1.2: Pressure and temperature changes throughout the freeze-drying process [3].
4
Chapter 2
Modelling The Freeze-Drying
Process
The freeze-drying method gained a lot of ground and is extensively used in other industries besides the pharmaceutical (e.g food industry). The high cost of the process and the
long times made the need for modeling imperative in order to gain better understanding
and identify the optimal operational conditions. These can be approached by different
methods. These are trial and error and later sequential analysis where the design of
an experiment is determined by the results of the previous. The approaches mentioned
give no answers as to the cause of the observed behaviors; therefore experiments must
be designed where the effect of altering single factors is explored [5]. Experiments on
the changes of single factors can be very time consuming however. As a result many
researchers used mathematical and computational modeling to tackle these problems.
Modeling lyophilization has several advantages and can have different uses depending
on its nature. The main benefit is that it can provide insight into the dynamics of freezedrying and thus enhance understanding of the underlying physics. Also, mathematical
and computational models can be used for both off-line and on-line optimization. Off-line
optimization can identify the optimal operational conditions while on-line can provide a
means to monitor the main parameters and make corrections in real time conditions.
Hence, these models help professionals make better decisions and can potentially reduce
both time and cost and improve product quality [5]. The nature of the model plays a crucial role as to how it can be used effectively. So, the form of the model is directly linked to
its purpose. There are complex 2D and 3D models that depend on many parameters and
can describe the phenomenon better and more realistically, but cannot be used for real
time applications or if they become too complex they might not even be able to enhance
understanding. Moreover, there are simplified models essential for routine application
[5] but there is the danger of oversimplification which could lead to results that are nor
realistic or useful. Finally, a basic factor as to the use of the model is whether or not its
predictions are important indicators of product quality [5]. Models whose predictions are
such, can be used for the development of tools of lyophilization cycles [5].
A number of models have been created to describe freeze-drying starting from very
simple and moving to more complex ones that take into consideration previously ignored
5
factors. The first models were 1D and described freeze-drying in trays whereas the later
models were 2D and described the process in vials, some taking into account the nature
of the vial and its position in a shelf of the freeze-drier.
2.1
Freeze-Drying in Trays
One of the first models describing the freeze-drying process was the non-steady state heat
and mass transfer model derived by Liapis & Litchfield [11]. It was used as an optimal
control model where, with the use of a quasi steady state analysis, they found some general operational policies for decreasing the process time. The control parameters of the
optimal control model were the chamber pressure and the radiator energy output. In
their1D model the frozen product of thickness L is placed on a tray where heating is provided from above. The equations describing the process are energy balances of the frozen
and dried regions, continuity equations and fluxes. A number of assumptions were made,
with the most characteristic for this model being that sublimation occurs parallel to the
surface of the sample and that the side and bottom are perfectly insulated. After the
derivation of the full model the conditions for optimality were derived using variational
calculus. In order to derive the analytical solution of these conditions they solved the
quasi-steady state version of the model. For the optimization part. four different cases
were considered based on two constraints. The system constraints were that the surface
temperature must not exceed a specific value called scorch temperature, Ts , of the dried
product and the interface must not exceed the melting temperature, Tm . Then, the four
cases were (i) neither constraint is active, (ii) surface temperature constraint active, (iii)
interface temperature constraint active and (iv) both constraints active. Their analysis
of the four cases indicated that initially both the pressure and the heat flux must be at
their maximum value until the interface constraint is reached where the pressure must be
dropped so as to keep the interface temperature at its maximum value without crossing
the critical temperature. Then, the heat input must also be reduced to keep the surface
temperature from exceeding the scorch temperature. To summarize, their work indicates
that an optimal operational policy is to run the process very close to the constraints,
which is something that will be found to be a very important policy in later modeling
papers. It is important to notice that in their model no bound water removal was considered, which occurs during primary drying, and that their results are found under the
assumption of quasi-steady state.
The two authors created a model based on the previous one with the same assumptions but including the removal of bound water during the primary drying stage [14].
In their improved model they allow for the occurrence of sublimation and desorption at
the same time. After deriving the equations the problem needed to be implemented numerically since an analytical solution was not possible. To this end, they used a specific
non-dimensionalization to immobilize the moving boundary and then solved numerically
using the Crank-Nicholson method. They simulated the drying of turkey meat and tested
the validity of their model against experimental data. Their results were in good agreement and predicted the drying time to acceptable accuracy. It is worth mentioning that
6
using these models, Litchfield, Liapis & Farhadpour checked the hypothesis that cyclical
pressure (periodically varying chamber pressure) can improve the drying times []. What
they found was that although that policy reduced the process time, an operational policy near an optimum value for the pressure gave consistently better results and required
minimum equipment [13],[12].
Moving on from models that only account for heating from the top, Millman et al. [16]
presented an 1D sorption-sublimation model where heat is provided independently both
from the top and the bottom plates and tested it against different operational policies to
find the one which produces the shortest time. In their model heat is also provided from
the sides but its effect is considered negligible with respect to the top and bottom and
is hence ignored and in contrast to the previous model, removal of free and bound water
does not occur simultaneously but one starts after the other has terminated. Moreover, a
very important aspect of their work was the development of bound water profiles which
is useful both for quality control and process termination. In this model the same assumptions and numerical methods were used as for the previous model. Four cases were
considered, each with each own plate conditions and with varying sample thickness and
chamber pressure and the simulation was conducted for skim milk. Skim milk contains
enzymes and proteins and hence it is very often used as it can be considered a complex
pharmaceutical product. Their results indicate that the policy which achieves the shortest overall time is the one that gives the shortest free water removal phase. Furthermore,
the largest heat and mass fluxes are achieved at policy C where the material constraints
are both active (Ts and Tm ). Finally, the sorbed water profiles produced from the model
are essential for predicting and minimizing quality deterioration [16].
In the context of freeze-drying in trays, there were two other models developed in
order to describe the dynamics of both drying stages (primary, secondary) for pharmaceuticals. The first model was the one developed by Liapis & Bruttini [10]. The purpose
of this model was to derive a mathematical model describing the dynamics of the drying
stages so as to analyze the drying rates which determine the drying time. Regarding the
formulation, heat is supplied from every direction but again the side heating is considered negligible and the assumptions remain the same. Two large sets of equations were
derived, that represent the two stages and the model was simulated and validated for
skim milk and cloxacillin monosodium salt which are both considered amorphous solutes.
The agreement with the experimental data was good for both stages and pointed towards
three rate controlling mechanisms for the removal of bound water in the secondary stage:
(i) desorption of bound water, (ii) mass transfer of water vapor by convection through the
dried material pores, and (iii) Knudsen and bulk diffusion of water vapor in the pores.
The second model, developed by Sadikoglu & Liapis [22], was based on the equations
and assumptions of the previous one. In the current formulation the dusty- gas model
was used to describe the transport of solvent vapor through the dried material pores.
The benefit of using this approach is that no details are needed for the structure of the
pores. A number of very interesting results were found from this model. The first is
that the effect of neglecting the removal of bound water during the primary stage did not
introduce a significant error. Moreover, as far as the mass transfer mechanisms are concerned, during the secondary drying stage, convective flow through the porous structure
7
was found to be insignificant for the determination of the dynamic behavior of this stage,
but they pointed out that this mechanism should not be ignored until further a number
of theoretical and experimental results suggested so.
2.2
Freeze-Drying in Vials
Following the development of 1D models describing freeze-drying in trays, multi-dimensional
models were developed that described freeze-drying in vials. These models offered a more
realistic representation of the process and were also concerned with a very important factor not present before, the position of the vial in the freeze-drier shelf.
The first 2D simulation of freeze-drying in vials was conducted by Mascarhenas et al.
[15]. The main limitations of 1D models addressed in this papers was the fact that rectangular materials were required and the interface had to be planar and parallel to the base,
hence not a curved shape. 2D mass and heat transfer phenomena were considered and the
interface could be of arbitrary shape and the arbitrary Lagrangian-Eulerian scheme was
used for tracking the sublimation interface. This scheme allows for the formulation of a
moving reference frame using a finite-element mesh consisting of grid points in arbitrary
spatial motion. The 2D mesh keeps changing throughout time to incorporate for the
thickness change of the frozen and dried region. As a result, the current model overcame
the limitations of 1D models and was used to describe both drying stages of skim milk
and BST. The results obtained were in accordance to previously published results and
provided information about the position and shape of the interface.
Sheehan & Liapis [24] constructed a 2D unsteady state model for the description of
the dynamic behavior of the two drying stages and tested it against different operational
policies. Regarding the previous model by Mascarhenas et al. [15] a number of problems
were identified and improved on in this paper. The main problem is that it fails to provide correct times for the drying stages because it does not account for cases were the
interface does not extent all the way along the diameter of the vials. A very important
feature of the model was the use of the side heat, which is the source of radial effects,
since it was pointed out that these effects are vital for correct 2D modeling. Moreover, a
reference was made to the difference in the condition of vials in different locations along
the freeze-drier shelves and they concluded that this phenomenon can be modeled with
the use of single vial models subjected to different, position dependent, conditions. In
their model primary and secondary drying coexist during the primary drying regime with
the first being mainly active in the centre, while the latter in the outer annular volume
element. Their results though indicated that the effect of desorption is negligible during
the primary stage, as was already pointed out in previously. For the numerical solution a
new set of variables was chosen so as to immobilize the moving boundary and the method
of orthogonal collocations was used. The equations were simulated for skim milk in three
cases. In the first case the vial is positioned in the outer array of vials on the shelf and
the plate temperature is kept at the highest allowed values. In the second case the vial is
again positioned in the outer array but with the temperature of the plate at a constant
value and finally in the third case the vial is in the centre with the same temperature
8
policy as the first case. After the completion of the primary stage the temperature is
set at the scorch temperature constraint. The shortest time was found for case I and
the slowest for case II. This result points towards a more aggressive policy close to the
temperature constraints. The authors noticed that a single temperature control profile
cannot be used for the whole array of vials without violating the constraints in some of
the vials, hence there is the need for modeling all vials as a whole and not separately for
different conditions. Finally, they concluded that aggressive control of the temperature,
near the constraints, offers faster times and more uniform temperature and bound water
distributions.
On 2002 Brulls & Rasmuson [6] created a theoretical model that systematically described heat transfer in vials. In their model the included effects, previously neglected,
such as heat accumulation in the vial, curvature of the vial bottom. Then, this heat
transfer model was used to develop a lyophilization model that was simulated to test the
influence of a number of factors (fullness, bottom curvature, position of the shelf, glass
vial) and compared against experimental data. This resulted in a very comprehensive
study of freeze-drying in vials. In their model the heat from the bottom of the vial was in
the form of radiation due to the curvature of the vial base, and it was observed that this
curvature had influence on the temperature that was correlated to pressure. For pressure
above a certain value this influence is significant but not so for pressure below that value.
This is very different to most of the previous models where the transfer of bottom heat
was through conduction. Also, different from other models is the fact that no heat was
considered from the top of the vials since their stopper acts as an insulator. Furthermore,
they noticed that there was a great difference between the shelf and vial temperature for
corner vials, due to heat transfer from other sources (sides), but no difference for the
center vials. Finally, the dynamics due to the shape of the vial cannot be ignored since
they found that heat accumulation plays a major role especially for low levels of fullness
which is relevant for pharmaceutical products [6].
As seen from previous models the position of the vial is very important and the radial effects introduced from the sides are not negligible. Later models, based on earlier
models, explored the effect of sides more systematically. Sadikoglu et al. [23] extended a
model discussed earlier [22] for freeze-drying in trays to include 2D effects and heat from
top bottom and sides and designed an optimal control model for freeze-drying and found
policies that reduced the total time markedly. What their results showed was that the
optimum result is obtained from running the process close to the temperature constraints
and keeping the pressure at its minimum values to increase water vapor flux, hence an
optimum value for the pressure. Also, very interesting was their finding that one of the
factors of optimum control is minimizing the heat transfer from the sides. This result
was further supported by Gan et al. [8] who used the model by Sheenan & Liapis [24]
to explore the effect of the tray sides in freeze-drier shelves. They tested three cases one
of which was with no sides. They found that this case gave the shortest times and the
most uniform distribution of bound water and of water loss. Their results also suggested
that the chamber walls should operate in a temperature slightly higher than the melting
temperature during primary stage and slightly lower than the scorch temperature during
secondary stage.
9
In order to optimize the primary drying stage of freeze-drying in vials, both with offline optimization and on-line monitoring and control, Velardi & Barresi [25] developed
a very detailed one-dimensional model. Due to the complexity and the high number of
equations this model was unsuitable for on-line use and therefore, two simplified models
were also developed based on the more complex one. Initially, they simulated vials placed
in the center of the vial array and thus shielded from side heating, using the detailed
model. The results were in good agreement with experimental data from freeze-drying
cycles. Then, having this model as a starting point they developed the two simplified
models. For the first model, heat and mass transfer was considered stationary due to
slow process and the heat was only supplied from the bottom. Also, heat transfer was
not considered in the dried layer. This model was not suitable for corner vials since it
assumed insulated sides. The second simplified model is more complex than the first
since heat transfer is considered for both the dried and frozen layers. Heat and mass
transfer are again stationary and heat from the environment is still ignored. It is very
interesting to notice that there was a very good agreement between the detailed and the
second simplified model. Both simplified models allowed for analytical solutions and their
simplified nature made them very useful for control and monitoring purposes and since
their first appearance they have been widely used for the development of model-based
tools for the freeze-drying cycles [5].
Going beyond the modeling of the two frying stages, there have been a few attempts
to model the freezing stage so as to improve the stage. Bruttini et al. [7] conducted
an exergy analysis and tested different policies to find the one that produces the least
exergy losses. Their results suggested that a distribution of temperatures of the cooling
source has significant benefits. Apart from reduction in exergy losses, a distribution of
temperatures can produce good freezing rates that form continuous and highly connected
crystals. A further modeling of the freezing stage was developed by Hottot et al. [9] who
used the commercial finite element code FEMLAB. The advantage of using FEMLAB
is that the real geometry of the vial can be drawn and used for real expressions of the
boundary conditions. Their model was in accordance to the experiments they conducted.
Concluding this chapter about modeling freeze-drying we saw that the overwhelming
majority of the models are concerned with the drying stages and little attention has
been paid to the freezing stage, despite its evident importance for the whole process [5].
The modeling of the drying stages seems to be very useful and serve its purpose by not
only providing insight into the process but also by guiding experiments and finding ways
of optimization and time reduction. Modeling in the field of freeze-drying is far from
complete as there are new technologies that can be used more effectively with the use of
models. Such technologies include alternative sources of energy to improve heat transfer
and dehydration rates (e.g. microwaves, infrared radiation, and radiofrequency), not yet
used for pharmaceutical due to product restrictions, and addition of biological solvent to
the water [5]. Finally, as already mentioned there is a great deal of unexplored modeling
space for the freezing stage.
10
Chapter 3
Mathematical Description of The
Small Time Behaviour of The
Freezing Stage in Vials
The main focus of this chapter is the modeling of the small time regime of the freezing stage in a 2D rectangular vial, using a ”Cauchy Integral Form” formulation of the
governing equation. A reason behind the interest in the small time approximation is to
check whether this formulation provides a valid basis for the complete solution of the
problem which is then derived using the same approach. For general times the problem
for the temperature between the vial walls and the interface, between the solid and liquid
states, is a Stefan (moving boundary) problem given by the heat equation along with the
appropriate non-dimensionalized boundary conditions.Thus
∇2 Θ(x, y, t) = β −1 ∂t Θ.
(3.1)
Where, β is the latent heat over the heat capacitance of the solidified phase. In this
case β −1 can be treated as a large number and the problem solved to first order, using
perturbation theory, as Laplaces equation ∇2 Θ(x, y, t) = 0. This approach is justified
and tested by Riley et al. [21], whose asymptotic solution was in good agreement to
numerical work that included the right hand side term of (3.1). Hence, in the remainder
of this work we shall consider Laplaces equation as the governing equation.
Here, Θ = 0 on the top and side walls (x = −1/2, x = +1/2, y = H), Θ = 1 on
the floor (y = 0) and in the liquid region, including the interface, and finally taking into
= (g2 +ggt2 )1/2 on the unknown interface given by g(x,y,t) =
account latent heat we have ∂Θ
∂n
x
y
0.
Our goal is to find the temperature inside the frozen region as well as the unknown
interface for small times. To this end we assume that for very small times the interface
will take a nearly rectangular form, similar to the vial, with very thin top and side layers
and with the bottom corners connecting to the vial’s walls, as shown in Figure 3.1, due
to boundary condition on the floor. The width of the vial is normalised to be equal to
unity with the origin situated in the centre of the floor.
11
Figure 3.1: Shape of the interface during freezing for small times and the boundary
conditions imposed on every side of the vial.
3.1
Top and Side Layers
Regarding the top and side layers, we assume that for small times they are very thin
(H −y << 1 for the top layer and x±1/2 << 1 for the side layers) and hence the 2D heat
equation becomes 1D since the y or x derivative dominates, to first order approximation,
in each case respectively. Furthermore scale analysis of the last boundary condition
indicates that the thickness of the layer is of O(t1/2 ) for t << 1. To see this consider the
top layer. Taking g ∼ y and Θ ∼ 1 the last boundary condition yields:
gt
∂Θ
= 2
,
∂n (gx + gy2 )1/2
1 y
∼ .
y t
Hence, the appropriate scaling for small times is y ∼ t1/2 . Using these simplifications
we derive below the equations for the temperature and the thickness of the layers.
For the top layer we define the thickness G as H-y. Considering the time scaling, G
and Θ can be expanded in orders of t1/2 to get:
12
G =t1/2 G0 + G1 + ...,
(3.2)
Θ =Θ0 + t1/2 Θ1 + ...
(3.3)
To get rid of t dependence a new variable is set as y*= y−H
. Hence, from (1.2) and
t1/2
from the definition of G we have y*= −G0 . The new equation and boundary conditions for
2
0
Θ0 = 0, Θ0 = 0 for y*= 0, Θ0 = 1 for y*= −G0 and ∂Θ
the temperature are ∂y∗
= − 12 G0
∂y∗
for y*= −G0 , all to leading order. The last boundary condition can be derived by taking
g to be equal to the leading order of G and noticing that the normal derivative w.r.t. to
y must have opposite sign to the normal derivative w.r.t. y*. From these it is trivial to
show that G0 = 21/2 and hence the equations for temperature and thichness are:
G =(2t)1/2 + G1 + ...,
y−H
+ t1/2 Θ1 + ...
Θ=−
(2t)1/2
Following exactly the same procedure but defining x* as x*=
tions are derived for the side layers with G0 equal to 21/2 again.
(3.4)
(3.5)
x±1/2
t1/2
the same equa-
The situation is more complicated in the case of the corners. In those regions the
heat equation must be solved for both y* and x*, of order unity, and it has to match
asymptotically the solutions for the thin layers. Also the interface shape is completely
unknown and cannot be taken as a constant to first order as before. Hence, a new
approach must be used to tackle the corners. To this end the problem will be formulated
using the Cauchy Integral Form so as to provide some equations that can be implemented
numerically/computationally to verify their validity.
3.2
Top Corners
The first step in modeling the freezing stage at the top corners of the vial for small times
is a conformal mapping from the initial plane, z = x*+iy*, to a new plane Z(X, Y ) = z 2 .
The advantage of this mapping is that the X axis is a known boundary (Θ = 0) and the
unknown interface is a curve on the upper half plane, as shown in Figure 3.2.
In this new plane the relation between the x*,y* and X,Y is given by the fillowing
equations:
X =x*2 − y*2 ,
Y =2x*y*.
13
(3.6)
(3.7)
(a)
(b)
Figure 3.2: (a) Right top corner of the interface situated in the lower quadrant in the
original z plane with x* and y* coordinates. (b) The interface corner after the mapping
to the Z plane with X and Y coordinates
As a result of the mapping the last boundary condition (latent heat) is subject to
∂Θ
, is not equal to ∂N
,
alterations. The first thing to consider is that the left hand side, ∂Θ
∂n
where N is the normal to the interface in the Z plane. Furthermore, in the rhs of the
equation the partial derivatives of g w.r.t. to x* and y* must be replaced with partial
derivatives w.r.t. X and Y. For the sake of clarity the full derivation can be found in
Appendix A. The resulting equation for the latent heat boundary condition in the new
plane is as follows:
XgX + Y gY
∂Θ
=−
.
2
∂N
4(X + Y 2 )1/2 (gX 2 + gY 2 )1/2
(3.8)
In order to use the Cauchy Integral form let us take a function Ψ which is related
to Θ by ΨY = ΘX . Taking the X derivative in both sides of this equation and using
the fact that Θ satisfies the heat equation we get ΨXY = −ΘY Y . Integrating w.r.t. Y
leads to the second relation ΨX = −ΘY . These two relations can be identified as the
Cauchy-Riemann equations and we can now define a function P(Z) such that, P(X,Y)
= Θ(X, Y ) + iΨ(X, Y ). Finally, by taking a closed contour in the Z plane, containing
the unknown interface Y = F(X) as seen in Figure 3.3, the Cauchy Integral along its
boundary represents the value of the function P(Z) inside that contour, where q = ξ + iη.
Z
P (q)
P (Z) =
dq.
(3.9)
C q−Z
14
Figure 3.3: Contour C in the complex Z plane containing the unknown inteface, the
known boundary on the X axis and the two sides, a distance b from the origin, where Θ
approaches asymptotically its solution for the top and side thin layer.
C can be broken down to four sub-contours. The first is the contour along the X-axis
where η = 0, dq=dξ and Ψ = Ψ0 (ξ). For the second and forth contours which are the side
contours, ξ = ±b, dq=dη and Θ, Ψ are functions of (± b, η). In this case these functions
will be represented by Θs1,2 , Ψs1,2 and their form is found using equation (3.5) in the
new coordinate system (X,Y) and the relation between Θ and Ψ. The third contour goes
dξ = dξ(1+iF 0 (ξ)).
along the unknown boundary where η = F (ξ) and hence dq = dξ +i dη
dξ
Also, on that contour Θ = 1 and Ψ = Ψ1 (ξ). In order to find the equations necessary for
the validation of this approach we take the imaginary part of (3.8).
1
2πi
Z
1
Im(
2πi
Z
Ψ(X, Y ) =Im(
Ψ(X, Y ) =
1
2π
Z
b
−b
b
iΨ0 (ξ)
1
dξ) + Im(
ξ − X − iY
2πi
−b
Z
b
−b
Z
0
F (b)
Θs1 (b, η) + iΨs1 (b, η)
dη)
b + η − X − iY
1 + iΨ1 (ξ)
1
dξ) + Im(
ξ + iF (ξ) − X − iY
2πi
iΨ0 (ξ)Y
1
dξ −
(ξ − X)2 + Y 2
2π
−b
1
+
2π
b
Z
0
F (b)
Z
0
F (−b)
Θs2 (−b, η) + iΨs2 (b, η)
dη)
−b + η − X − iY
Θs1 ∗ (η − Y ) − Ψs1 ∗ (b − X)
dη
(b − X)2 + (η − Y )2
(ξ − X)(1 − F 0 Ψ1 ) + (F − Y )(F 0 + Ψ1 )
1
dξ +
2
2
(ξ − X) + (F − Y )
2π
Z
0
F (−b)
Θs2 ∗ (η − Y ) + Ψs2 ∗ (b + X)
dη
(b + X)2 + (η − Y )2
The final step towards deriving the desired equations is using the previous result and
considering two cases. In the first case Y→ 0 and Ψ(X, Y ) → Ψ0 (X). In the second case
15
Y→ F (X) and Ψ(X, Y ) → Ψ1 (X). Caution is needed when treating the integrals in these
two cases so as to find the correct leading order contribution. So, taking the two limits
and using the final boundary condition, which using the Cauchy-Riemann equations can
∂Θ
= − ∂Ψ
, we arrive at a system of three equations for three unknowns
be expressed as ∂N
∂S
(F, Ψ1 , Ψ0 ). The partial derivative of Ψ w.r.t. the tangential direction, S, can be easily
rewritten as a partial derivative w.r.t. X from the fact that dS 2 = dX 2 + dY 2 . A full
derivation of Θs and Ψs as well as the special treatment of the integrals involved in the
two above mentioned limit situations can be found in Appendix B,C respectively.
1
Ψ1 (X) = F 0 (X)+ F (X)
π
Z
b
−b
Ψ0
1
dξ −
2
2
(ξ − X) + F (X)
π
Z
F (b)
0
Θs1 ∗ (η − F ) − Ψs1 ∗ (b − X)
dη,
(b − X)2 + (η − F (X))2
Z
Z
1
1 − F 0 Ψ1
1 F (b) Θs2 ∗ (η − F ) + Ψs2 ∗ (b + X)
+ − b
dη.
dξ +
π −b ξ − X
π 0
(b + X)2 + (η − F (X))2
1
Ψ0 (X) =
π
Z
b
−b
Z
(ξ − X)(1 − F 0 Ψ1 ) + F (F 0 + Ψ1 )
1 F (b) Θs1 η − Ψs1 ∗ (b − X)
dξ −
dη,
(ξ − X)2 + F (ξ)2
π 0
(b − X)2 + η 2
Z
1 F (b) Θs2 η + Ψs2 ∗ (b + X)
+
dη.
(3.11)
π 0
(b + X)2 + η 2
Ψ01 (X) =
3.3
(3.10)
XF 0 (X) − F (X)
.
4(X 2 + F (X)2 )1/2
(3.12)
Bottom Corners
In order to derive the equations for the top corners we follow the same steps, regarding
the function P(z), but a conformal mapping is not needed in this case since the problem
is simple enough. Taking as an example the left bottom corner, there are a few differences
to the top corner. The contour is obviously different and now the interface g(x*,y*)=0
will be given by g(x*,y*) = x* - f(y*), hence on the interface x*=f(y*). The new contour,
c, can be broken into three parts. The first part is the vertical line h → 0 for x*=0,
where dq=dη, Θ = 0 and Ψ = Ψ0 . The second part consists of the interface and here
dq = dη(i+f(η)), Θ = 1 and Ψ = Ψ1 . The last part is the horizontal line from f(h) →
0 where dq=dξ, and Θs , Ψs are functions of (ξ,h). Taking again the imaginary part for
the two limit cases, x*→ 0 and x*→ f(y*), as well as the latent heat boundary condition,
we get the following three governing equations, their full derivation is very similar to the
one found in Appendix B for the top conrers and will be omitted:
1
Ψ1 (y*) = −f (y*) +
π
0
√
+
1
π
Z
0
h
Z
0
2
Z √2
Ψ0 f (y*)
1
1 + f 0 Ψ1
dη
+
−
dη
f (y*)2 + (η − y*)2
π 0
η − y*
Θs ∗ (ξ − f ) + Ψs ∗ (h − y*)
dξ
(ξ − f )2 + (h − y*)2
16
(3.13)
√
1
Ψ0 (y*) =
π
Z
2
0
1
f (Ψ1 − f 0 ) + (η − y*)(1 + f 0 Ψ1 )
dη +
2
2
f + (η − y*)
π
√
Z
2
0
Θs ξ − Ψs ∗ (h − y*)
dξ (3.14)
ξ 2 + (h − y*)2
1
Ψ01 (y*) = − (f (y*) + f 0 (y*)y*)
2
(3.15)
Figure 3.4: Contour c in the complex z plane containing the uknown inteface, the known
boundary on the y* axis and top, a distance h from the origin, where Θ approaches
asymptotically its solution side thin layer.
3.4
Full Model
The above equations for the corners can be tested by implementing them numerically
and constructing an iterative process by making an initial guess for the interface function
(F,f) and then recalculating it using the expressions for Ψ1 and Ψ0 . This will was not
conducted for this project but it can provide a good starting point for a later project.
To derive the corresponding equation for the full problem for the time O(1) the same
procedure is followed. A contour is defined at the z-plane, Figure (3.5), and taking the
imaginary part of the complex function P(x,y) and the limits y→ 0 and y→ f (x, t) gives
the three equations. The goal is to find an expression for the unknown (latent heat)
boundary condition for Θ. The three equations are given below, following a derivation
almost identical to the previous equations.
(β)
Ψ0 (x, t)
1
=−
π
Z
1/2
−1/2
1
(1 − f 0 Ψ1 )(ξ − x) + (Ψ1 + f 0 )f
dξ
−
(ξ − x)2 + f 2
π
1
+
π
Z
0
−H
Z
0
−H
(α)
Ψ0 η
dη,
(1/2 − x)2 + η 2
(γ)
Ψ0 η
dη.
(1/2 + x)2 + η 2
17
(3.16)
Z
Z
(α)
1 0
1 1/2 1 − f 0 Ψ1
Ψ0 (η − f )
ξ−
Ψ1 (x, t) = f (x, t) − −
dη,
π −1/2 ξ − x
pi −H 1/2 − x)2 + (η − f )2
Z 1/2
Z
(β)
(γ)
Ψ0
Ψ0 (η − f )
1
1 0
− f (x, t)
dξ
+
dη.
(3.17)
2
2
π
π −H 1/2 + x)2 + (η − f )2
−1/2 (ξ − x) + f
0
Finally, using the initial latent heat boundary condition (for the non-scaled coordinates) and the relation between Θ and Ψ, as was used before, we get:
∂Ψ1 (x, t)
= ft (x, t).
∂x
(3.18)
Solving this system of equations could potentially give us the unknown boundary
condition on the interface and hence allow for the full solution of the temperature using the
heat equation (3.1). An analytical solution is not possible and thus this has to be carefully
implemented numerically taking into account the fact that there are Principal value
integrals. This approach seems promising and once it has been validated by numerical
calculation it can be used to predict the temperature distribution during the freezing
stage, any asymmetries of the process as well as the total time, all very vital information.
Figure 3.5: Contour for the full problem in the complex z plane containing the uknown
inteface f and the known boundary along the walls of the vial.
18
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20
Appendix A
Latent Heat Boundary Condition
After Conformal Mapping
As mentioned before the derivation of the last boundary condition in the new plane, Z,
consists of two parts. The rhs of the original equation needs to be rewritten w.r.t. X,Y,
the new variables, and the partial derivative of Θ w.r.t. the normal on the interface is
not the same in both planes. To this end let us first consider the partial derivatives of gc ,
where gc is the interface function in the corners. By the use of the chain rule we have:
∂gc ∂X
∂gc ∂Y
∂gc
∂gc
∂gc
=
+
= 2x*
+ 2y*
∂x*
∂X ∂x* ∂Y ∂x*
∂X
∂Y
(A.1)
∂gc
∂gc ∂X
∂gc ∂Y
∂gc
∂gc
=
+
= −2x*
+ 2y*
∂y*
∂X ∂y* ∂Y ∂y*
∂Y
∂X
(A.2)
By substituting (A1) and (A2) in the original rhs we get:
2
c
c
c
− 2y*2 ∂g
+ 2x*y* ∂g
1 2x* ∂gc + 2x*y* ∂g
∂Y
∂X
∂Y
rhs = − q ∂X
,
2 4(x*2 + y*2 )( ∂gc )2 + 4(x*2 + y*2 )( ∂gc )2
∂X
∂Y
1
(x*2 − y*2 )gc X + 2x*y*gc Y
p
=− q
,
2
2
2
2
g
+
g
c
c
X
Y
2 x* + y*
rhs = −
2(X 2
1
Xgc X + Y gc Y
p
.
2
1/4
+Y )
gc 2X + gc 2Y
(A.3)
The difference in the normal derivative after the conformal mapping can be found by
making again use the chain rule.
21
∂X
dx* +
∂x*
∂y
dY =
dx* +
∂x*
dX =
∂X
dy* = 2x*dx* − 2y*dy∗
∂y*
∂y
dy* = 2y*dx* + 2x*dy∗
∂y*
As a result,
dX 2 + dY 2 = 4(x*2 + y*2 )(dx*2 + dy*2 )
Finally we know that dN 2 = dX 2 + dY 2 and dn2 = dx*2 + dy*2 and so we get,
dN
dn = √
4 X2 + Y 2
(A.4)
From (A3) and (A4) we can now write down the complete boundary condition in the
Z plane as seen in the main text.
Xgc X + Y gc Y
∂Θ
=−
.
2
∂N
4(X + Y 2 )1/2 (gc X 2 + gc Y 2 )1/2
22
(A.5)
Appendix B
Leading Order Approximation of
Integrals In The Two Limit Cases
Taking the imaginary part in the equation for P(Z) we are left with an equation for
Ψ(X, Y ) that involves integrals on a contour. As our goal was to find specific equation for
Ψ on the boundaries, two limits were taken, Y → 0 and Y → F(X). When considering these
limiting cases caution is needed in order to find the correct leading term. The integrals
involved in the in these equation can be divided into two, more general, categories.
Z ∞
H(ξ)(ξ − X)
dξ,
(B.1)
2
2
−∞ (ξ − X) + (Y − F (X))
Z ∞
H(ξ)(Y − F (X)
dξ.
(B.2)
2
2
−∞ (ξ − X) + (Y − F (X))
From the above general forms it is clear that the case where Y → 0 is not a problem.
This is not the case for Y → F(Y) where the effect is different for the two integrals.
The first integral has a logarithmic dependence on X. It can be broken down into three
integrals:
Z X−
Z X+
Z ∞
H(ξ)
H(ξ)
H(ξ)
dξ +
dξ +
dξ.
−∞ ξ − X
X− ξ − X
X+ ξ − X
Where << 1. Taking the limit → 0 the middle integral becomes zero and the the rest
two are the definition of a Principal value integral where the contribution comes from
H(ξ) throught the integral and is hence global.
Z ∞
H(ξ)
−
dξ.
(B.3)
−∞ ξ − X
For the second form the main contribution comes from ξ → X since there is no logarithmic dependence any more on X and the numerator goes to 0 as Y → F. Consequently,
to leading order, H can be taken out of the integral as H(X) along with Y-F and the integral can be evaluated.
23
Z
∞
H(X)(Y − F (X))
(ξ −
−∞
1
H(X)(Y − F )
Y −F
H(X)[tan−1 (
Z
∞
−∞
X)2
1
dξ,
+ (Y − F (X))2
1
dξ,
( Yξ−X
)2 + 1
−F
ξ−X ∞
)] ,
Y − F −∞
H(X)π.
Hence the leading term.
24
Appendix C
Asymptotic Limits of Θ, Ψ On The
Corner’s Sides
In the small time regime the values for Θ, Ψ on the corners must approach asymptotically
their values in the top and side thin layers. Having derived the equation for Θ for these
regions, Ψ can be found using the Cauchy-Riemann equation in the z plane. For the top
layer is used to give the equation for Ψ where Θ = − √y*2 as found.
∂Ψ
1
x*
= √ =⇒ Ψ = √ .
∂x*
2
2
For the side layers where Θ is given by Θ = − √x*2 we use the second relation, Ψy* = Θx* .
∂Ψ
1
y*
= √ =⇒ Ψ = √ .
∂y*
2
2
In order to find the equation for Θ and Ψ in the Z plane we need to use the equations
connecting X,Y with x*,y* but we need to be careful with so as to chose the correct sign.
By solving the system of algebraic equation (1.6-1.7) w.r.t. y* we arrive at two possible
solutions:
p
√
−X + X 2 + Y 2
√
.
y* = ±
2
To chose the correct sign we need the notice that y* is negative since its sign is
controlled by y-H which starts from 0 and goes to -H for increasing time. Hence, the
equation for the asymptotic solution of Θ in the Z plane as X → ∞ is given by:
p
√
−X + X 2 + Y 2
Θs1 =
.
(C.1)
2
The relevant equation for X → −∞ is given by solving for x* initially or we can guess
it intuitively since there is symmetry in the Z plane along the Y axis and hence Θ must
be even.
p
√
X + X2 + Y 2
Θs2 =
.
(C.2)
2
25
The same holds for Ψ which can be found again using the Cauchy-Riemann equations.
Ψs1
Ψs2
p
√
X + X2 + Y 2
=−
,
2
p
√
−X + X 2 + Y 2
=−
.
2
26
(C.3)
(C.4)